# Picards Existence and Uniqueness theorem doesn't prove anything

• Al3105
In summary: This theorem is just as important as the Picard's theorem, but it may not be as well-known because it is a generalization of it. In summary, the Picard-Lindelöf theorem states that if the right-hand side of an initial value problem in an ordinary differential equation is continuous, then the solution exists and is unique on the entire interval on which the differential equation is defined. This theorem is a generalization of Picard's Existence and Uniqueness theorem and is just as important, but may not be as well-known.
Al3105
As far as I can understand it, Picard's Existence and Uniqueness of ODEs theorem relies on the fact that a the given function f(x,t) in the initial value problem dx/dt = f(x,t) x(t0) = x0 is Lipschitz continuous and bounded on a rectangular region of the plane that it's defined on. And the theorem(mainly through successive approximations) proves that the solution exists and is unique ONLY in that particular region.

I believe that there exists a theorem which proves, that the Picard theorem implies existence and uniqueness for the whole Interval that the given function is defined on. However I can not for the life of me find that theorem.

So how does this work? Am I correct in what I stated above? If I am, then why isn't that theorem as easy to find as Picard's theorem? It would seem that it's just as important.

Could someone explain the theorem and perhaps give or help me find a proof for it?

The Picard–Lindelöf theorem (also called the Picard–Lindelöf existence and uniqueness theorem) states that if a given initial value problem in an ordinary differential equation has a continuous right-hand side, i.e., f(x,t) is continuous for all (x,t) in a rectangular region of the x-t plane, then it has a unique solution. This theorem is a generalization of the Picard's Existence and Uniqueness theorem which requires that f(x,t) be Lipschitz continuous on a rectangular region of the plane. The proof of this theorem follows from the fact that if a function is continuous, then it is locally Lipschitz continuous, and thus satisfies the conditions of the Picard's Existence and Uniqueness theorem. As such, the solution to the differential equation exists and is unique in any subregion of the rectangular region of the x-t plane, and hence, by the principle of analytic continuation, also exists and is unique in the entire rectangular region. In other words, Picard–Lindelöf theorem states that if the right-hand side of an initial value problem in an ordinary differential equation is continuous, then the solution exists and is unique on the entire interval on which the differential equation is defined.

## 1. What is Picard's Existence and Uniqueness theorem?

Picard's Existence and Uniqueness theorem is a mathematical theorem that guarantees the existence and uniqueness of solutions to certain types of differential equations. It is named after the French mathematician Emile Picard.

## 2. How does the theorem work?

The theorem states that if certain conditions are met, such as continuity and differentiability of the functions involved, then there exists a unique solution to the differential equation in a given interval. This solution can be found using the Picard iteration method.

## 3. What does it mean when people say the theorem "doesn't prove anything"?

When people say that the theorem "doesn't prove anything," they are often referring to the fact that it does not provide a constructive method for finding the solution to a differential equation. Instead, it only guarantees that a solution exists and is unique.

## 4. Are there any limitations to the theorem?

Yes, there are limitations to the theorem. It only applies to certain types of differential equations, and the conditions required for the theorem to hold can be quite restrictive. Additionally, the theorem does not work for differential equations with non-continuous or non-differentiable functions.

## 5. How is Picard's Existence and Uniqueness theorem useful in science?

The theorem is useful in science because it provides a rigorous mathematical framework for studying and solving differential equations. It allows scientists to prove the existence and uniqueness of solutions to these equations, which are often used to model real-world phenomena in fields such as physics, engineering, and biology.

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